There's already a question like this here so that my question could be considered duplicate, but I'll try to make my point clear that this is a different question.

Is there a way to derive Biot-Savart law from the Lorentz' Force law or just from Maxwell's Equations?

The point is that we usually define, based on experiments, that the force felt by a moving charge on the presence of a magnetic field is $\mathbf {F} = q\mathbf{v}\times \mathbf{B}$, but in that case the magnetic field is usually left to be defined later.

Now can that force law be used in some way to obtain Biot-Savart law like we obtain the equation for the electric field directly from Coulomb's Force law?

I wanted to know that because as pointed out in the question I've mentioned, although Maxwell's Equations can be considered more fundamental, those equations are obtained after we know Coulomb's and Biot-Savart's laws, so if we start with Maxwell's Equations to obtain Biot-Savart's having use it to find Maxwell's Equations then I think we'll fall into a circular argument.

In that case, without recoursing to Maxwell's Equations the only way to obtain Biot-Savart's law is through observations or can it be derived somehow?

  • $\begingroup$ Neither Maxwell nor Biot-Savart are fundamental - all such formulas follow from Coulomb and a well-chosen definition of $B$, as mentioned tangentially in this short rant. $\endgroup$ – user10851 Jun 8 '13 at 20:13
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    $\begingroup$ @ChrisWhite, Maxwell's equations do not follow just from the Coulomb law, special relativity and definitions. For example, the Gauss law for non-rectilinear motion of charges cannot be derived without further assumptions. $\endgroup$ – Ján Lalinský Feb 7 '15 at 12:38
  • $\begingroup$ I think @Hans de Vries can provide an elegant answer. $\endgroup$ – Ayan Biswas Apr 10 '16 at 16:42

$\def\VA{{\bf A}} \def\VB{{\bf B}} \def\VJ{{\bf J}} \def\VE{{\bf E}} \def\vr{{\bf r}}$The Biot-Savart law is a consequence of Maxwell's equations.

We assume Maxwell's equations and choose the Coulomb gauge, $\nabla\cdot\VA = 0$. Then $$\nabla\times\VB = \nabla\times(\nabla\times\VA) = \nabla(\nabla\cdot\VA) - \nabla^2\VA = -\nabla^2\VA.$$ But $$\nabla\times\VB - \frac{1}{c^2}\frac{\partial\VE}{\partial t} = \mu_0 \VJ.$$ In the steady state this implies $$\nabla^2\VA = -\mu_0 \VJ.$$ Thus, we have Poisson's equation for each component of the above equation. The solution is $$\VA(\vr) = \frac{\mu_0}{4\pi}\int \frac{\VJ(\vr')}{|\vr-\vr'|}d^3 r'.$$ Now we need only calculate $\VB = \nabla\times\VA$. But $$\nabla\times\frac{\VJ(\vr')}{|\vr-\vr'|} = \frac{\VJ(\vr')\times(\vr-\vr')}{|\vr-\vr'|^3}$$ and so $$\VB(\vr) = \frac{\mu_0}{4\pi}\int \frac{\VJ(\vr')\times(\vr-\vr')}{|\vr-\vr'|^3} d^3 r'.$$ This is the Biot-Savart law for a wire of finite thickness. For a thin wire this reduces to $$\VB(\vr) = \frac{\mu_0}{4\pi}\int \frac{I d{\bf l}\times(\vr-\vr')}{|\vr-\vr'|^3}.$$

Addendum: In mathematics and science it is important to keep in mind the distinction between the historical and the logical development of a subject. Knowing the history of a subject can be useful to get a sense of the personalities involved and sometimes to develop an intuition about the subject. The logical presentation of the subject is the way practitioners think about it. It encapsulates the main ideas in the most complete and simple fashion. From this standpoint, electromagnetism is the study of Maxwell's equations and the Lorentz force law. Everything else is secondary, including the Biot-Savart law.

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    $\begingroup$ But how I've seen it done, Maxwell's equations are derived from the biot-savart law, which would make this circular. $\endgroup$ – JLA Jun 8 '13 at 20:24
  • $\begingroup$ @JLA: I've added something to address the "circularity" you refer to. $\endgroup$ – user26872 Jun 8 '13 at 23:00
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    $\begingroup$ @JLA, it is not possible to mathematically derive Maxwell's equations from the Biot Savart law. What people sometimes do is to infer (arrive at) Maxwell's equations from the Biot-Savart law for a specific case like stationary currents and then generalize them to all situations by word. $\endgroup$ – Ján Lalinský Feb 7 '15 at 12:40
  • $\begingroup$ For the sake of clarity, differential operators are applied on ${\bf r}$ and not ${\bf r'}$, that's how they are swapped with integrals over ${\bf r'}$. $\endgroup$ – A.G. Aug 30 '17 at 11:53
  • $\begingroup$ @A.G. Indeed, taking the derivative with respect to ${\bf r'}$ makes no sense. We have $\nabla = \sum \hat e_i \partial/\partial x_i$, not $\sum \hat e_i \partial/\partial x'_i$ (for which I would write $\nabla'$ or some such thing). $\endgroup$ – user26872 Sep 9 '17 at 18:32

It may be true that in days of yore people measured the force resulting from a filamentary current, discovering the Biot-Savart law, and then in turn used that as inspiration to construct Maxwell's equations. If that's how it actually happened historically, fine.

But this is analogous to some alien archaeologist 10 million years from now finding a skeletal hand and foot in the Earth. From the hand, the archaeologist comes to understand what the animal who had that hand liked to do with it: that it could grasp and use tools and so on. From the foot, the archaeologist it comes to understand that the animal it belonged to walked on two legs and that it typically weighed in adulthood around 100-300 pounds.

Only later does the archaeologist that the hand and the foot both belonged to the same animal--a human being. But the nature of the work means that the puzzle of what a human being was has to be broken down into chunks that can be individually understood before the whole picture can come together. That said, it would be backwards to suggest the hand and the foot are more fundamental than the human being itself.

The Maxwell equations have been constructed to be consistent with the Biot-Savart law and other pieces of information, like Coulomb's law. Thus, you can derive Biot-Savart from Maxwell, but not the other way around, for Maxwell is more general and all-encompassing.

If you already know the Lorentz force law, you can infer the strength of the magnetic field from a wire just by shooting charged test particles near the wire and observing their motion. But this calls into question how you already know the Lorentz force law, and so on.

You can go in circles all day over what is or is not fundamental, over what must be based on experimental observation and what is merely constructed to be consistent with those observations, but often there's a preference for "simple" experimental observations being considered fundamental vs. theoretical constructs that incorporate many such observations--see Chris White's comment that Maxwell's equations can be derived from Coulomb's law and some other stuff.

To me, this is silly. Maxwell's equations incorporate the sum total of our observations (those that fit the classical regime, at least). To me, it is what we know about classical electromagnetism. To say that you can derive Maxwell's equation with only one result plus a few assumptions...well, it misses the point that those assumptions also had to be tested and verified in the first place. To me, it is very very backwards to single out special cases (pure electric, pure magnetic, static or dynamic fields) and treat them as "fundamental".

Edit: but really, a physicist needs to work in both directions. To create new theory, we often have special cases that we don't know are connected and must bridge them together. That's building Maxwell's equations from Coulomb's law and Biot-Savart. To analyze a particular problem most easily, one we're not sure there is a special-case formula for, we must resort to the most general description (Maxwell) and try to reduce it down to something simpler and easier to solve (in the case of no current and no time-dependence, you can get back to Coulomb's law). Both approaches are necessary to be as flexible as possible.

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Starting from a Rowland Ring type experiment it is possible to define permeability as a measure of the flux generated in a unit volume per ampere-turn. If we then assume this flux to dissipate as an inverse square law we obtain the biot savart law as a magetic analogue of coulomb's law with the addition of the cross product taking care of the perpendicularity of the field direction and strictly on the understanding that it is a working hypothesis validated by its usefulness since a current element cannot exist in isolation from the rest of its circuit. My advice - Ignore all temptations to fall into more maths than the minimum needed, that will lead you to understanding. Hope this helps.

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Kindly follow the following link.https://en.wikipedia.org/wiki/Jean-Baptiste_Biot and plese go to the heading "Work". It says that the law was discovered experimentally in in the year 1820 i.e. 45 years before the Maxwell equations were published. The general formulation to the Biot-Savart law was given by P. Laplace. The expression of the Biot-Savart Law (the integration) shows that the principle of superposition is already included in it. Maxwell equations were developed later and they were designed suitably to encompass the implications of the Biot-Savart's law. Perhaps that is the reason why we can derive Maxwell's equations from Biot-Savart's law and vice versa.

Please go to this link https://en.wikipedia.org/wiki/Lorentz_force and go to the "History" section. It says that in the year 1881 i.e. 16 years after the Maxwell equations were published, Thomson first derived a form of Lorentz force law form Maxwell equations. Finally the modern form of Lorentz force law was derived by Lorentz in 1892 from the Maxwell equations.

So the Historical sequence is like this:

Biot-Savart's Law ==> Maxwell's Equations ==> Lorentz force law.

But in the classrooms we are taught in the following sequence:

First: The Lorentz force law, to introduce the concept that magnetic field exerts force on a moving charge.

Second: The Biot-Savart law, to introduce the concept that moving charges produce magnetic field.

Third: The Maxwell's equations; the generalization of all the experimental observations in electromagnetism.

So the conclusion is:

(1) Biot-Savart's law is an experimentally observed law. This law also includes the idea that superposition principle is also valid in magnetostatics. This law provided the foundation for magnetostatics.

(2) Maxwell's equations were derived is such a way as to encompass the findings of Biot-Savart's law (along with other experimental observations of electromagnetism). It's a theoretical generalization. Maxwell's equations are more fundamental than any other experimental observation because experiments usually are done under certain circumstances and thus can not give a generalized information.

(3) Lorentz force law was derived from Maxwell's equations but can be directly verified experimentally.


"Observation and then generalization": I think this is the way physics is developed. Observation (experiment) always establishes the foundation. Generalization encompasses the observation and extends it usability to other imaginable configurations, cases and circumstances. Hence it is always possible to derive generalization from observation and vice versa [Biot-Savart law can be derived from Maxwell equations and Maxwell equations can be derived from Biot-Savart Law].

Here it is emphasized that Biot-Savart's Law is the important observation which started the field of magnetostatics. Maxwell equations (Generalization) and concept of vector potential (a general property of vector field) can be used to derive Biot-Savart's law but that does not signify that the Law is just an intermediate step in development of knowledge regarding magnetostatics. That it is possible to derive Biot-Savart's law form Maxwell equations and the concept of vector potential only certifies that the Generalization in Maxwell equations is correct.

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  • $\begingroup$ But the OP was not asking about the historical order of events. $\endgroup$ – Danu May 5 '16 at 12:52

We have to look at the time-line (the history). Biot-Savart law was published prior to the publication of Maxwell Equations. So it is Gauss’s Law for Magnetic Fields (the Second Maxwell Equation) which is derived from Biot-Savart Law and not the other way around. The derivation of Gauss’s Law for Magnetic Fields (the Second Maxwell Equation) from Biot-Savart Law can be read here Gauss’s Law for Magnetic Fields

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The problem with Biot-Savart's Law is that theoretically it is formulated in terms of current elements $Idl$ and then integrated. But in most textbooks it is formulated also for POINT charges, in terms of $qv$. The problem here is that when a point charge $q$ moves with velocity $v$ the magnetic field in nearby spaces CHANGES with time, ie, we have a $\frac{dB}{dt}$, and then induction effects take place and the magnetostatic condition is violated. In contrast when $Idl$ is integrated along a continuous wire the $B$ field is constant, (magnetostatic). The two situations are very different and to the best of my knowledge the point charge $B$ field has never been measured directly. The Force on $qv$, yes, but not the field produced by $qv$.

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